A104144 a(n) = Sum_{k=1..9} a(n-k); a(8) = 1, a(n) = 0 for n < 8.
0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144, 16272, 32512, 64960, 129792, 259328, 518145, 1035269, 2068498, 4132920, 8257696, 16499120, 32965728, 65866496, 131603200, 262947072, 525375999, 1049716729, 2097364960
Offset: 0
Links
- T. D. Noe, Table of n, a(n) for n = 0..208
- Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
- Taras Goy and Mark Shattuck, Some Toeplitz-Hessenberg Determinant Identities for the Tetranacci Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.6.8.
- Kai Wang, Identities for generalized enneanacci numbers, Generalized Fibonacci Sequences (2020).
- Eric Weisstein's World of Mathematics, Fibonacci n-Step Number
- Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1,1,1).
Crossrefs
Programs
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Maple
for n from 0 to 50 do k(n):=sum((-1)^i*binomial(n-8-9*i,i)*2^(n-8-10*i),i=0..floor((n-8)/10))-sum((-1)^i*binomial(n-9-9*i,i)*2^(n-9-10*i),i=0..floor((n-9)/10)):od:seq(k(n),n=0..50);a:=taylor((z^8-z^9)/(1-2*z+z^(10)),z=0,51);for p from 0 to 50 do j(p):=coeff(a,z,p):od :seq(j(p),p=0..50); # Richard Choulet, Feb 22 2010
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Mathematica
a={1, 0, 0, 0, 0, 0, 0, 0, 0}; Table[s=Plus@@a; a=RotateLeft[a]; a[[ -1]]=s, {n, 50}] LinearRecurrence[{1, 1, 1, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 1}, 50] (* Vladimir Joseph Stephan Orlovsky, May 25 2011 *) With[{nn=9},LinearRecurrence[Table[1,{nn}],Join[Table[0,{nn-1}],{1}],50]] (* Harvey P. Dale, Aug 17 2013 *) -
PARI
a(n)=([0,1,0,0,0,0,0,0,0; 0,0,1,0,0,0,0,0,0; 0,0,0,1,0,0,0,0,0; 0,0,0,0,1,0,0,0,0; 0,0,0,0,0,1,0,0,0; 0,0,0,0,0,0,1,0,0; 0,0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,0,1; 1,1,1,1,1,1,1,1,1]^n*[0;0;0;0;0;0;0;0;1])[1,1] \\ Charles R Greathouse IV, Jun 16 2015
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PARI
A104144(n,m=9)=(matrix(m,m,i,j,j==i+1||i==m)^n)[1,m] \\ M. F. Hasler, Apr 22 2018
Formula
a(n) = Sum_{k=1..9} a(n-k) for n > 8, a(8) = 1, a(n) = 0 for n=0..7.
G.f.: x^8/(1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9). - N. J. A. Sloane, Dec 04 2011
Another form of the g.f. f: f(z) = (z^8-z^9)/(1-2*z+z^(10)), then a(n) = Sum_((-1)^i*binomial(n-8-9*i,i)*2^(n-8-10*i), i=0..floor((n-8)/10))-Sum_((-1)^i*binomial(n-9-9*i,i)*2^(n-9-10*i), i=0..floor((n-9)/10)) with Sum_(alpha(i), i=m..n)=0 for m>n. - Richard Choulet, Feb 22 2010
From N. J. A. Sloane, Dec 04 2011: (Start)
Let b be the smallest root (in magnitude) of g(x) := 1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9, b = 0.50049311828655225605926845999420216157202861343888...
Let c = -b^8/g'(b) = 0.00099310812055463178382193226558248643030626601288701...
Then a(n) is the nearest integer to c/b^n. (End)
Extensions
Edited by N. J. A. Sloane, Aug 15 2006 and Nov 11 2006
Incorrect formula deleted by N. J. A. Sloane, Dec 04 2011
Name edited by M. F. Hasler, Apr 22 2018
Comments